The chaotic hypothesis is an extension of the ergodic hypothesis to mechanical systems in stationary states but out of equilibrium (e.g. an ionized gas in an electric field which generates a stationary current). The hypothesis is that such systems behave "essentially" as uniformly hyperbolic systems. This is an idea that goes back to the early '70's (Ruelle) and which recently has produced some concrete "predictions" and is therefore worth pushing to the extreme. The talk will illustrate the few successes and many problems and difficulties that have been generated by the attempts to understand better the chaotic hypothesis.